Invariants
| Base field: | $\F_{61}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 25 x + 271 x^{2} - 1525 x^{3} + 3721 x^{4}$ |
| Frobenius angles: | $\pm0.0746782053317$, $\pm0.283932848163$ |
| Angle rank: | $2$ (numerical) |
| Number field: | \(\Q(\sqrt{-322 +50 \sqrt{29}})\) |
| Galois group: | $D_{4}$ |
| Jacobians: | $5$ |
| Cyclic group of points: | yes |
This isogeny class is simple and geometrically simple, primitive, ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.
Newton polygon
This isogeny class is ordinary.
| $p$-rank: | $2$ |
| Slopes: | $[0, 0, 1, 1]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $2443$ | $13541549$ | $51548582575$ | $191740222452149$ | $713336156254883728$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $37$ | $3639$ | $227107$ | $13848219$ | $844588302$ | $51520029663$ | $3142739546347$ | $191707303124019$ | $11694146241606517$ | $713342914323716854$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 5 curves (of which all are hyperelliptic):
- $y^2=31 x^6+14 x^5+19 x^4+53 x^3+50 x^2+48 x+39$
- $y^2=23 x^6+4 x^5+15 x^4+44 x^3+46 x^2+35 x+6$
- $y^2=47 x^6+42 x^5+46 x^4+46 x^3+59 x^2+36 x+23$
- $y^2=29 x^6+55 x^5+50 x^4+44 x^3+31 x^2+55 x+33$
- $y^2=6 x^6+5 x^5+18 x^4+35 x^3+6 x^2+19 x+35$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{61}$.
Endomorphism algebra over $\F_{61}$| The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-322 +50 \sqrt{29}})\). |
Base change
This is a primitive isogeny class.
Twists
Below is a list of all twists of this isogeny class.
| Twist | Extension degree | Common base change |
|---|---|---|
| 2.61.z_kl | $2$ | (not in LMFDB) |